Can I make a force field that is defined with differential equations?
As the documentation says,


  You can use a Texture force field to create an arbitrarily complicated force field, which force in the three directions is color coded. Red is coding for the X-axis, green for the Y-axis and blue for the Z-axis (like the color of the coordinate axes in the 3D View). A value of 0.5 means no force, a value larger than 0.5 acceleration in negative axis direction (like -Z), a value smaller than 0.5 acceleration in positive axis direction (like +Z).


You can create the texture corresponding to your equations using Mathematica or similar packages. Or you could write your own program to output the texture.